Anisotropic scaling and generalized conformal invariance at Lifshitz points

4 Pages·0.84 MB·English

by  Malte Henkel

Anisotropicscalingandgeneralized conformalinvarianceat Lifshitz
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0 points
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MalteHenkela,1 andMichel Pleimlinga,b

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aLaboratoire de Physique des Mat´eriaux2, Universit´e HenriPoincar´e Nancy I, B.P. 239,
a
J F–54506 Vandœuvre–l`es–Nancy Cedex, France
bInstitut fu¨rTheoretische Physik 1, Universit¨atErlangen-Nu¨rnberg, D–91058 Erlangen, Germany
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v Abstract
4
5 AnewvariantoftheWolffclusteralgorithmisproposedforsimulatingsystemswithcompetinginteractions.This
4 method is used in a high-precision study of the Lifshitz point of the 3D ANNNI model. At the Lifshitz point,
8 several critical exponentsare found and theanisotropic scaling of thecorrelators is verified.The functional form
0 of thetwo-point correlators is shown tobeconsistent with thepredictions of generalized conformal invariance.
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/ Keywords: Conformalinvariance, Lifshitzpoint, ANNNImodel, correlationfunction, Wolff algorithm
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a PACS: 05.70.Jk, 64.60.Fr, 02.70.Lq, 11.25.Hf
m
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d
n Competing interactions are encountered in a previous Monte Carlo studies of the transition(s)
o large variety of physical systems such as, among fromthedisorderedhigh-temperaturephasetothe
c others,magnets,alloysor ferroelectrics.These in- ordered low-temperature phases exclusively used
:
v teractionsmayleadtorichphasediagramswitha single-spin flip algorithms [4,5]. Due to the criti-
i multitudeofphasesaswellastospecialmulticrit- calslowing-downinherenttothesemethods,those
X
ical points called Lifshitz points (LP). At an LP, numericalstudieswerelimitedtorathersmallsys-
r
a a disordered, a uniformly ordered and a periodi- tem sizes. However, large systems are needed for
cally ordered phase become indistinguishable [1]. theprecisecomputationofcriticalquantities,such
A large number of systems (magnets, ferroelec- ascriticalexponentsorcriticalcorrelators.
tric liquid crystals, uniaxial ferroelectrics, block In this contribution, we present a new variant
copolymers)havebeen showntopossesanLP. of the Wolff cluster algorithm [6] specifically de-
This kind of systems may be mimicked by spin signedforthe simulationofsystems withcompet-
modelswithcompetinginteractions.Thesimplest ing interactions near criticality. Similarly, we also
of these models is the well-known ANNNI (ax- generalizetherecentlyproposedmethodofEvertz
ial next nearest neighbour Ising) model [2,3]. Be- and von der Linden for the computation of corre-
causeofthepresenceofthecompetinginteractions, lation functions of infinite systems [7] to systems
with competing interactions. As an example, we
1 Correspondingauthor: [email protected] present results obtained at the LP of the three-
2 Laboratoireassoci´e auCNRSUMR7556
Preprintsubmitted toComputer Physics Communications 1February2008
+ − − + + + − − − +
dimensional ANNNI model, and we shall concen-
(a) − + − + − − + + + −
trateontheLPcriticalexponentsandcriticalcor-
relators. In particular, our data for the scaling of + + + − + − − + + +
the two-point correlation functions are consistent + − − + + − + − − −
withthetheoreticalpredictionsofgeneralizedcon-
− − − − + + + − − +
l
formalinvariance[8,9]. a
− + − + + + − − − + i
Problemscomingfromcriticalslowing-downen- x
+ + + − + + + − + + a
countered when using local Monte-Carlo dynam-
icsarealleviatedbyusingnon-localmethods,such − − − + − + − − − +
as the Wolff cluster algorithm [6]. For the Ising + + + + + − + + + +
model with only a nearest-neighbour coupling J,
+ + − + + + − − + −
this algorithm may be described as follows: one
choosesrandomlyalatticesite,theseed,andthen
builds up iteratively a cluster. If i is a cluster site
+ − − + + + − − − +
(withspins )asitej neighbouringtheclustersite
i (b) − + − + − − + + + −
i will be included into the cluster with probabil-
ity p = 1(1+sign(s s ))(1−exp[−2J/(k T)]). + + + − + − − + + +
2 i j B
One ends up with a cluster ofspins having allthe + − − + + − + − − −
same sign, see Fig. 1a, and which is flipped as a − − − − + + + − − +
l
whole.Thiskindofsame-signclustersisobviously a
− + − + + + − − − + i
not adapted to our problem because of the com- x
+ + + − + + + − + + a
peting interactions.
The modified cluster algorithm we propose for − − − + − + − − − +
simulating spin models with competing interac- + + + + + − + + + +
tions is in the following presented for the ANNNI + + − + + + − − + −
model.Ageneralizationtoothermodelsisstraight-
forward.TheHamiltonianofthe3DANNNImodel Fig. 1. Typical clusters (gray spins) obtained (a) by the
is,withs =±1, Wolff algorithm and (b) by our modified algorithm for
xyz
systems with competing interactions, here shown in two
dimensions for simplicity. The competition takes place in
H=−JXsxyz(cid:0)s(x+1)yz+sx(y+1)z+sxy(z+1)(cid:1) theaxialdirection.Thespinenclosedbyacircleistheseed.
xyz
whole,containsspinsofbothsigns,asshowninFig.
+κJXsxyzsxy(z+2) (1) 1b.AtvariancewiththetraditionalWolffmethod,
xyz
the flipped spins are not necessarily connected by
whereas J > 0 and κ > 0 are coupling constants. nearest-neighbour couplings. Ergodicity and de-
Intheaxialz-directioncompetitionbetweenferro- tailedbalanceareprovenasusual.Thisalgorithm
magneticnearest-neighbourandantiferromagnetic works extremely well in the paramagnetic phase,
next-nearest-neighbour couplings occurs, leading in the ferromagnetic phase and in the vicinity of
to arichphase diagram[10]. the LP.It has not yet been subjected to stringent
Our proposed modified cluster algorithmstarts tests in the modulated regionwhere largefree en-
with a randomly chosen seed and builds up iter- ergy barriers make simulations very difficult (we
atively a cluster. Consider a newly added cluster pointoutrecentprogressachievedinmicrocanoni-
latticesiteiwithspins .Alatticesitej withspin calsimulationsoffiniteANNNIsystems[11]).Note
i
s nearest neighbour to i is included with proba- thatouralgorithmdiffersfromthesame-signclus-
j
bilityp =p,whereasanaxialnext-nearestneigh- ter algorithm proposed by Luijten and Blo¨te [12]
n
boursitekwithspins isincludedwithprobability for the simulation of systems with long-range in-
k
p = 1(1−sign(s s ))(1−exp[−2Jκ/(k T)]). teractions.
a 2 i k B
Thus the final cluster, which will be flipped as a Inaninteresting work,Evertzandvonder Lin-
2
den [7] have proposed a new numerical scheme, Table1
EstimatedlocationoftheLPinthe3DANNNImodelde-
basedontheWolffclusteralgorithm,forthedirect
finedonacubiclattice,asobtainedfromhigh-temperature
computation of two-point functions of an infinite
expansions (HT)and Monte Carlosimulations (MC).
lattice. In their approach, each cluster is started
at the same lattice site and not from a randomly κL TL
chosensite.Afteracertainnumberofclusterflips, HT[13] 0.270 0.005 3.73 0.03
± ±
the two-pointfunctions canbe measuredwithin a
MC [5] 0.265 3.77 0.02
certain region, this region getting larger with in- ±
presentwork0.270 0.0043.7475 0.005
creasingnumberofflips.Withthismethod,corre- ± ±
lationfunctionsofthe infinitelattice areobtained
Table2
attemperaturesabovethecriticaltemperature[7].
Criticalexponents atthe LPof the3D ANNNImodel, as
At the critical temperature, finite-size effects are
obtained from Monte Carlo simulations (MC) and renor-
greatly reduced as compared to more traditional malized field theory (FT). The numbers in brackets give
approaches. We have generalized the method of theestimated errorinthe lastdigit(s).
[7] to systems with competing interactions by re-
α β γ (2 α)/γ β/γ
placingthe usualWolffclusterswithourmodified −
clusters.Thisenablesustocomputewithunprece- MC [5] 0.19(2) 1.40(6) 0.14(2)
dentedprecisiontwo-pointcorrelatorsinspinsys- FT [14] 1.27 0.134
tems withcompeting interactions.
FT [17] 0.160 0.220 1.399 1.315 0.157
In the following, we report results of a large-
scale Monte Carlo study of the uniaxial LP en- presentwork0.18(2)0.238(5)1.36(3) 1.34(5) 0.175(8)
counteredinthe3DANNNImodelwherethenew
algorithms introduced above have been used with in good approximationequal to 1/2 [14]. In order
greatsuccess[9].Thewholestudytooktheequiva- to take into account the special finite-size effects
lentofmorethantwoCPU-yearsonaDECalpha comingfromtheanisotropicscalingattheLP[15],
workstation. For the investigation of LP proper- largesystemsofanisotropicshapewithL×L×N
ties a precise location of this point is mandatory. spins with 20 ≤ L ≤ 240 and 10 ≤ N ≤ 100 have
Onepossibilitytolocatethispointisgivenbythe beensimulated.
structure factor S(q) = PhS0Sri exp(iqz) as the Our estimates for the LP exponents α,β,γ are
r givenin Table 2. For the first time, these ANNNI
transition between uniformly orderedand period-
model exponents are computed independently.
ically ordered phases shows up as a shift of the
They were found by investigating effective expo-
maximum of S(q) from q = 0 to a non-zero value
nents which yield the critical exponents in the
[4]. Having found a reliable value for κ , the LP
L limit T −→ T [9,16] (provided that finite-size
critical temperature is then obtained by standard L
effects canbe neglected). Ourerrorbars takeinto
methods. In Table 1, we compare the location of
account both the sample averaging and the un-
the LP as obtained from our data with previous
certaintyinthe locationofthe LP.The reliability
estimations.NotethatKaskiandSelke[5]didnot
of our data is illustrated by the agreement of the
attempt an independent determination of κ but
L independently estimated exponents α, β and γ
merely determined T for a κ in close vicinity to
L with the scaling relation α + 2β + γ = 2 up to
κ as obtained from a high-temperature (HT) se-
L ≈0.8%.Furthermore,theagreementwitharecent
ries expansion [13]. From Table 1 the increase in
two-loopcalculation[17]isremarkable.
precisionis evident.
Fig.2summarizesourdataobtainedfortheLP
The uniaxial LP is a strong anisotropic critical
point where the correlation lengths parallel and spin-spin correlatorC(r⊥,rk) = hSr⊥,rkS0⊥,0i. It
showsthescalingfunctionΦ(u)relatedtothecor-
perpendicular to the axial direction diverge with
relatorby
different critical exponents: ξk,⊥ ∼ |T −TL|−νk,⊥.
Thevalueoftheanisotropyexponentθ =νk/ν⊥is C(r ,r )=r−2x/θΦ(rθ/r ) (2)
⊥ k k ⊥ k
3
usedintheinsetofFig.2.Theniceagreementbe-
tween theory and numerical data provides strong
1
1
u) evidence for the applicability of the generalized
Φ( conformal invariance to the strongly anisotropic
0.5 criticalbehaviourrealizedatLPs.Asimilaragree-
)
u ment between numerics and theory has been ob-
(
Φ
tainedfor the energy-energycorrelationfunction.
0.5
0
0 1 2 3 4 5 6 7 Inconclusion,wehavepresentedanewvariantof
1/u1/2
theWolffclusteralgorithmdesignedforthe study
ofsystemswithcompetinginteractions.Appliedto
the LP of the 3D ANNNI model, the anisotropic
0 scaling of the two-point correlators could be con-
0 1 2 3
u1/2 firmed directly for the first time. The form of the
scalingfunctionisinagreementwiththehypothe-
sisofgeneralizedconformalinvariance.
Fig. 2. Scaling funtion Φ(u) versus u1/2 = (√r⊥/rk)1/2
for κ=0.270 and T =3.7475. The different symbols cor-
respondtodifferentvaluesofr⊥.Inset:comparisonofthe References
full data set of 1.7 104 points for the scaling function
×
Φ(u)(graypoints)withtheanalyticalpredictionresulting
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field-theoreticalcalculations[14,17]arenotyetdis-
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d3Ω(v) dΩ(v) 2x
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1
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Themeanvalueofpobtainedinthiswayhasbeen
4